Statistical mechanics and stability of a model eco-system

Abstract

Abstract. We study a model ecosystem by means of dynamical techniques from disordered systems theory. The model describes a set of species subject to competitive interactions through a background of resources, which they feed upon. Additionally direct competitive or co-operative interaction between species may occur through a random coupling matrix. We compute the order parameters of the system in a fixed point regime, and identify the onset of instability and compute the phase diagram. We focus on the effects of variability of resources, direct interaction between species, co-operation pressure and dilution on the stability and the diversity of the ecosystem. It is shown that resources can be exploited optimally only in absence of co-operation pressure or direct interaction between species.

Analytical theory of species abundance distributions of a random community model

“…There is also a consideration on a more realistic model not only with random interactions but also with resource competition (Yoshino et al 2007) where we studied stability conditions as a function of parameters, e.g., the number of resources and the variance of resource quantities.…”

“…8, Lyapunov function f in the symmetric case or L p ðxÞ in the antisymmetric case is not known and we cannot use the methods of equilibrium statistical mechanics such as the replica method (Mezard et al 1987;Nishimori 2001)in the case of symmetric interactions. There are, however, some techniques of nonequilibrium statistical physics using the path integral or the generating functional to analyze the global behavior of a system with random interactions (Rieger 1989;Opper and Diederich 1992;Galla 2005Galla , 2006Yoshino et al 2007Yoshino et al , 2008. Such techniques have been recently applied to cross-disciplinary areas such as neural networks (Gardner et al 1987;Rieger et al 1988;Düring et al 1998;Katayama and Horiguchi 2001;Mimura et al 2004), and the minority game in economics and social science (Challet et al 2005;Coolen 2005).…”

“…where À1 c 1 is the level of symmetry the same as in Eq. 8 (Opper and Diederich 1992;Galla 2006;Yoshino et al 2007Yoshino et al , 2008. The scaling factor 1=N is introduced for the same reason in Eq.…”

“…Parameter u([0) is productivity which is same as in the case of symmetric interactions in Eq. 12 and is also termed ''cooperation pressure'' because diversity (the number of coexisting species) is in general a monotonically increasing function of u (Diederich and Opper 1989;Opper and Diederich 1992;Tokita 2004;Yoshino et al 2007Yoshino et al , 2008. We also here use f ðtÞ ð1=NÞ P N j¼1 P N k¼1 a jk x j x k for the average fitness.…”

Characterizing Species Distributions by Productivity and Mortality Rates in Multispecies Models

“…The implications of open systems and environmental noise are discussed later, but not addressed specifically in this paper. Equation (1) has been examined by Tokita and Yasutomi [2003], Tokita [2004], Yoshino et al [2007] and others. In their analyses, they use "replicator equations" (Hofbauer and Sigmund [1998]), whereby the state variables of equation (1) are normalized.…”